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Department of Physics, Chemistry and Biology

Master’s Thesis

Few-Particle Effects in Semiconductor Quantum Dots:

Spectrum Calculations on

Neutral and Charged Exciton Complexes

Kuang-Yu Chang

August 14, 2010

LITH-IFM-A-EX--10/2362—SE

Linköpings universitet Institutionen för fysik, kemi och biologi

581 83 Linköping

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Departmant of Physics, Chemistry and Biology

Few-Particle Effects in Semiconductor Quantum Dots:

Spectrum Calculations on

Neutral and Charged Exciton Complexes

Kuang-Yu Chang

August 14, 2010

Supervisor

Fredik Karlsson, IFM

Examiner

Per-Olof Holtz, IFM

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Abstract

It is very interesting to probe the rotational symmetry of semiconductor quantum dots for quan-tum information and quanquan-tum computation applications. We studied the effects of rotational symmetry in semiconductor quantum dots using configuration interaction calculation. More-over, to compare with the experimental data, we studied the effects of hidden symmetry. The 2D single-band model and the 3D single-band model were used to generate the single-particle states. How the spectra affected by the breaking of hidden symmetry and rotational symmetry are discussed.

The breaking of hidden symmetry splits the degeneracy of electron-hole single-triplet and triplet-singlet states, which can be clearly seen from the spectra. The breaking of rotational symmetry redistributes the weight percentage, due to the splitting of pxand py states, and gives a small brightness to the dark transition, giving rise to asymmetry peaks. The asymmetry peaks of 4X, 5X, and 6X were analyzed numerically. In addition, Auger-like satellites of biexciton recombination were found in the calculation. There is an asymmetry peak of the biexciton Auger-like satellite for the 2D single-band model while no such asymmetry peak occurs for the 3D single-band model. Few-particle effects are needed in order to determine the energy separation of the biexciton main peak and the Auger-like satellite.

From the experiments, it was confirmed that the lower emission energy peak of X2− spec-trum is split. The competed splitting of the X2− spectra were revealed when temperature dependence was implemented. However, since the splitting is small, we suggest the X2− peaks are broadened in comparison with other configurations according to single-band models. Fur-thermore, the calculated excitonic emission patterns were compared with experiments. The 2D single-band model fails to give the correct energy order of the peaks for the few-particle spectra; on the other hand the peaks order from 3D single-band model consistent with experimental data.

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Acknowledgments

I gratefully thank my supervisor, my teacher, and my friend Fredrik Karlsson, who not only guided me through out this work and gave me countless advices, but also broadened my knowl-edge of solid state physics and to model real world. Your intuitive sense to the calculated spectra and your instant e-mail responses make this work incredibly efficient! I sincerely appreciate all those time that you spent on answering my endless questions. Moreover, your comments on the experimental spectra help me a lot to link the modeled results with the physical understanding. Most importantly, I enjoy being your student.

I am also grateful to Arvid Larsson, who introduced me to the µ-PL lab. I know nothing in the lab, but your introduction and explanation make things crystal clear. I have learned a lot during the lab times with you.

I would also like to thank my girl friend Chorpure Thinprakong and my roommate Yu-Te Hsu and for all the fun time together and lots of valuable discussions. Further more, my gratitude goes to my family for their supports.

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Preface

Background

It has been a great triumph for the semiconductor industry during the last several decades. By making the microchip smaller and smaller, it finally gave rise to nanotechnology. Thanks to the developments of nanotechnology, it is now possible to fabricate nanostructured semicon-ductor with high quality, namely quantum wells (QWs), quantum wires (QWRs), and quantum dots (QDs). Electrons in these nanostructured devices are quantum confined. Many exciting quantum mechanical effects have been observed, such as quantum Hall effect [1]. From the ap-plication point of view, these low-dimensional semiconductor devices have been very interesting for their great potentials in various fields. To be specific, nanosized quantum dots may be used as single photon source, emitting photon one by one, for applications in quantum communica-tion. Moreover, symmetric quantum dot may be used as the source for entangled photon pairs and correlated photons for quantum computation.

Besides quantum mechanical interests, III-V semiconductors have attracted many atten-tions. GaAs, AlAs, InP, and their ternary alloy, like AlxGa1−xAs, have been important in many applications. Different from silicon (VI group) based semiconductors, two general prop-erties for most of III-V semiconductors are the high carrier mobility and the direct band gap. This reason makes III-V semiconductor a better choice for optical device when compared to Si semiconductor.

To combine the optical property of III-V semiconductors and quantum confinement effects of nanosized heterostructured semiconductors, the object of this thesis work is the spectra cal-culation of pyramidal AlGaAs/GaAs/InGaAs quantum dots, shown in Figure 1. These specific quantum dots are grown by metal organic chemical vapor deposition (MOCVD) technique with highly controlled geometry [2].

AlGaAs

InGaAs

QD

∼µ

m

Figure 1: Side view of heterostructured AlGaAs/GaAs/InGaAs pyramidal quantum dot. Self-formed quantum dot is located at the top of micrometer-sized pyramid. [3]

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PREFACE

Motivations

Since the perfectly symmetric quantum dot is the ideal source for entangled photon pairs [4], it is very interesting to probe the symmetry of quantum dots, in the perspective of quan-tum computation and quanquan-tum optics applications. By controlling the excitation power of photoluminescence, the number of trapped charge carriers in quantum dots can be controlled [5]. Together with effective mass approximation of semiconductor materials (section 2.1.1) and configuration interaction (CI) (chapter 2.2.2), so called few-carrier system can be solved in reasonable time frame without further approximation [6, 7].

Many works have devoted to the spectra prediction with different number of charge carrier [5, 8, 9]. There are two symmetries are usually assumed. One is the symmetry between electron wave function and hole wave function, sometimes referred as hidden symmetry [10]. The other is the rotational symmetry of the quantum dot structure.

However, some features from the experimental data have not yet been explained. We at-tempt to solve those unexplained mysteries by the breaking of quantum dot symmetries. There-fore, the present thesis is focused on the asymmetry effects on the optical spectra, where the asymmetries are introduced to the confinement potential of quantum dots.

Methods

The calculation procedure can be summarized as follow:

1. Calculate single-particle states according to the selected quantum dot model. It was done by solving the single-particle Schr¨odinger equation using finite difference equation. (Appendix B).

2. Generate the CI Hamiltonian for few-particle systems (section 2.2.2). The matrix elements were calculated by Fourier convolution. Only Coulomb interactions were considered. 3. Diagonalize the CI Hamiltonians for each few-particle system. The diagonalization

pro-cedures were done by Matlab commend eig.

4. Generate the spectra according to Fermi golden rule.

All the codes were written in Matlab script. And the calculations were done on Green server of Linkoping University, which is a 74 nodes Linux cluster system. Detail flow diagrams are shown in Appendix D.

Thesis plan

chapter 1 introduces on the physical properties and some essential concepts of semicon-ductor and quantum dots. Including the energy band, crystal structure, and the concept of quantum dots.

chapter 2 establishes the theoretical background to describe the charge carriers in the quantum dots, which is the path to few-particle effects. Including the effective mass approx-imation, optical transition, configuration interaction, second quantization formalism, and the dipole operator for optical transition.

chapter 3 describes two models, 2D single-band model and 3D single-band model, that were used in this thesis work. Some results of single-particle calculation are shown.

chapter 4 discusses the calculated results. Includes the neutral exciton spectra and the charged exciton spectra. We focused on the asymmetry effects on the emission spectra.

chapter 5 presents some of the measured spectra of the few-particle configurations, which were obtained from the experiments.

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Contents

Abstract iii

Acknowledgments iv

Preface v

1 Introduction to Semiconductors and Quantum Dots 1

1.1 Electron in Solid . . . 1

1.2 Energy Bands . . . 3

1.3 Crystal Structure . . . 4

1.4 Quantum Dots and Excitons . . . 5

2 Single- and Few-Particle States 7 2.1 Single-Particle States . . . 7

2.1.1 Effective Mass Approximation for Quantum Dots . . . 7

2.1.2 Optical Transitions . . . 8

2.2 Few-Particle States . . . 8

2.2.1 Notations for Few-Particle System . . . 8

2.2.2 Configuration Interaction . . . 9

2.2.3 Second Quantization Formalism . . . 9

2.2.4 Few-Particle Hamiltonian . . . 10

2.2.5 Dipole Operator . . . 11

3 QD Models and Symmetries 12 3.1 2D Single-Band Model . . . 12

3.2 3D Single-Band Model . . . 14

3.3 Symmetries of Quantum Dots . . . 17

4 Results and Discussions 19 4.1 Notations and Concepts . . . 19

4.2 Neutral Excitons . . . 20

4.2.1 Breaking of Hidden Symmetry . . . 20

4.2.2 Breaking of Rotational Asymmetry . . . 22

4.2.3 Auger-like Satellites: Biexcition . . . 27

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CONTENTS

4.3.1 X2− Spectra and Temperature Dependence . . . . 30

4.3.2 Negatively Charged Excitons . . . 33

5 Experiment: Few-particle Spectra 35 6 Summary 37 A Theoretical Derivations 38 A.1 Effective Mass Approximation for QDs . . . 38

A.2 Fermi Golden Rule . . . 39

A.3 Lagrange Minimization for CI . . . 41

A.4 One- and Two-Particle Operators . . . 41

B Finite Difference Equation 44 B.1 1D Case . . . 44

B.2 2D and 3D Cases . . . 45

C Coulomb Matrix Elements 47 C.1 Coulomb Integral . . . 47

C.2 Comparison with Analytical Results . . . 47

C.3 Tables of Numerical Values . . . 47

D Flow Diagrams 52

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Chapter 1

Introduction to Semiconductors and

Quantum Dots

This chapter introduces some general but essential concepts of semiconductors and quantum dots. We start with the description of the Bloch electrons, which follows by the introductions of energy band and the crystal structure. Then, we introduce the quantum dots and the notations for exciton complexes in quantum dots.

1.1

Electron in Solid

Knowledge of electrons is necessary to understand the properties of material. Not only electric and thermal properties are determined by the behavior of electrons, but also the structure of crystal and molecules, optical properties are depended on the electronic structure [11].

Free electron

To start with, consider the free electron, which is not being attracted by any potential. The Hamiltonian to describe such electron is ˆHF ree = p

2

2me, where p = −i~∇ is the momentum operator. The eigenstates of this Hamiltonian are plane waves, ψk(r) = eik·r. Instead of quantum number (n, l, m) in atomic physics, wave vector1 k is a good quantum number to identify the energy of each state, namely the dispersion relation

E(k) = ~

2 2me

|k|2. (1.1)

One obtains the empty lattice band structure from Equation (1.1) for any type of crystal structure [12].

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Chapter 1. Introduction to Semiconductors and Quantum Dots

Bloch function

In a solid, The full Hamiltonian can be written as follow: ˆ H = P i p2 i 2mi + P j P2j

2Mj kinetic energies of electrons and nuclei +12 P

i,i0 e2 4π0|ri−ri0|

electron-electron Coulomb repulsions

+12 P j,j0

ZjZj0e2

4π0|Rj−Rj0| nucleus-nucleus Coulomb repulsions

−P

i,j

Zje2

4π0|Rj−ri|, electron-nucleus Coulomb attractions

(1.2)

where summation index i runs though all the electrons while index j runs though all the nuclei. pi and ri are the momentum operator and position operator of ith electron, while Pj and Rj are the operators for nuclei. Zj is the atomic number of jth nucleus [13].

However, it is possible to neglect the kinetic energy of the nuclei for the present purpose. The nucleus-nucleus interactions therefore become a constant under such approximation. Further more, those moving electrons tend to screen out the embedded positive nuclei by the factor e−ksr, where ks is the Thomas-Fermi screening length [14]. As a result, it is possible to neglect the electron-nucleus attraction terms and rewrite the electron-electron interaction terms as some periodically deformed constant, which is the effective potential of the crystal [15], Vcry(r). That is, we have the periodicity

Vcry(r + T) = Vcry(r),

where T is the crystal translation vector. Combine all these approximations together, Equation (1.2) takes the form of the Schr¨odinger equation with a periodic potential, namely

ˆ H ≈ ˆHper = X i p2 i 2mi + Vcry(r), which can be solved analytically,

ˆ Hper(r)ψ(r) =  − ~ 2 2me ∇2+ V cry(r)  ψ(r) = Eψ(r). (1.3)

The solution is, the Bloch function,

ψ(r) = uk(r)eik·r. (1.4)

where uk(r) has the same periodicity as Vper(r), i.e uk(r + T) = uk(r) [14]. Electron described by Equation (1.4) is a Bloch electron. It is the fundamental tool to describe electrons in solid. Similar to free electron case, energy for each state can be expressed by k. Therefore, the energy near band edge2 can be approximated as

( Ec(k) = Eg + ~ 2 2m∗ e|k| 2, Ev(k) = − ~ 2 2m∗h|k| 2, (1.5)

where m∗e and m∗h are the effective masses (see section 2.1.1) of Bloch electrons and holes ,while Ec and Ev is the energy of conduction band and valence band respectively. For simplicity, Bloch electron is abbreviated as electron though out this thesis, unless otherwise noted.

2From the derivation of nearly free electron model [14], one gets E(k) = ∆E + (1 ± C)k2~2

2me, where C is some

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1.2 Energy Bands

1.2

Energy Bands

As shown in Figure 1.1, when two atoms come closer to each other, the energy level splits into bonding state, |Atom1i + |Atom2i, and antibonding state, |Atom1i − |Atom2i, e.g. |si → |s1i ± |s2i or |pi → |p1i ± |p2i, where s and p represent s-orbital and p-orbital respectively. For

-5 0 5 -4 -2 0 Ener gy (ar bitr ar y unit) -5 0 5 -4 -2 0 -5 0 5 -4 -2 0

distance (arbitrary unit)

(a) (b) (c)

bonding antibonding

bonding antibonding

atomic Coulomb potential

Figure 1.1: Numerical illustrations of bonding state and antibonding state. Arbitrary units of distance and energy are used to illustrate the splitting of energy levels as two atoms come closer to each other. Atomic distance between two atom is decreasing from (a) to (c).

the case of GaAs, there are three valence electrons from Ga (4s2 4p1) and five valence electrons from As (4s2 4p3). It is sufficient to consider four valence electrons per atom (4s2 4p2), with slightly deformed crystal potential. The situation is similar in the case of AlxGa1−xAs. We may plot the energy levels of four valence electrons as functions of atomic distance (Figure 1.2(a)). Note that the filled orbitals of valence electrons are two bonding state, |s1i + |s2i and |p1i + |p2i. This result can be extended to the case with more atoms, namely solid (Figure 1.2(b)). Single energy level splits into energy band3. The filled and the empty orbitals become the valence band and the conduction band [12].

Conduction Band

Excited states in the conduction band are the excited electrons. For the present purpose, we concern only the electrons near the lower band edge. As shown in Figure 1.2(b), the conduction band at the lower band edge is rooted from empty |s1i − |s2i orbital. It is important to point out the S-like property at the lower band edge of conduction band. The spherical symmetry of S-like orbital makes the structure of conduction band much simpler than valence band.

Valence Band

Excited states in valence band are the vacancies of electrons. It can be considered as the quasi-particles, holes, which sometimes referred as hole picture. In the hole picture, holes in valence band are positively charged particles with opposite spins of their electron counter parts. Again, we concern only the holes near the upper edge of valence band. As shown in Figure 1.2(b), valence band at the upper band edge is rooted from the filled |p1i − |p2i orbital in Figure 1.2(a). There are two things to be considered. The first is the non-isotropic nature of p orbitals, e.g. px, py, and pz, which give rise to heavy hole and light hole. The second is the spin-orbit coupling. Since the total angular momentum of p-states is nonzero, spin-orbit coupling, Hso ∝ S · L, is also nonzero. The spin-orbit term slightly deforms all the bands and gives rise to another split-off band. Figure 1.2(c) shows the schematic band structure of

3Only direct band gap are considered in this section because the object of present thesis is direct band gap

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Chapter 1. Introduction to Semiconductors and Quantum Dots Atomic distance Ener gy s1+s2 s1-s2 p1+p2 p1-p 2 Lattice constant Ener gy a0 filled orbitals empty orbitals valence band conduction band p s p s (a) (b) Eg electrons Energy k heavy holes light holes split-off holes (c) Eg

Figure 1.2: Schematic illustration of energy level vary with atomic distance, using tightly bound electron model. (a) Two atoms case, energy levels split as atomic distance decreases from infinite. Dash lines are empty orbitals and solid lines are filled orbitals. (b) Solid case, energy levels split into energy bands (gray region). Band gap and lattice constant are shown as Eg and a0 [12]. (c)

Schematic illustration of the band structure near Γ point. Electron band, heavy hole band, light hole band, and split-off band are shown. [13].

conduction band and valence bands. However, a more sophisticated theory, e.g k · p theory, is needed for detailed and quantitative explanations.

The hole picture will be used to describe valence band though out this thesis, unless other-wise noted. Furthermore, electrons and holes are sometimes referred as charge carriers.

1.3

Crystal Structure

(111)

group III elements (Al, Ga, In) group V element (As)

[100] a0

[010]

Figure 1.3: Schematically illustrate the zinc-blende structure of III-V semiconductor [3]. Prin-cipal crystallographic direction [100] and [010] are shown, as well as the crystal plane (111).

Knowledge of crystal structure is necessary to understand the quantum dots. The crystal structure of GaAs is zinc-blende structure. It can be constructed by diamond structure with two different elements, as shown in Figure 1.3. Each group III element is surrounded by four group V element and vise versa. For the case of ternary alloy, like AlxGa1−xAs or InxGa1−xAs, Aluminum (indium) takes the place of Ga randomly according to the mole fraction x. One important consequence of the ternary alloy is that the band gap depends on the mole fraction. Figure 1.4 shows the relations between band gap and molar fraction for AlxGa1−xAs.

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1.4 Quantum Dots and Excitons

x

valence band (eV)

conduction band (eV)

0.5 0.0 0.5 1.0 x ~ 0.45 0.0 -0.5 0 1 GaAs AlAs Γ L X X L Γ 1.424 eV

direct band gap

indirect band gap

Figure 1.4: Plot of band gap versus molar fraction, x, for AlxGa1−xAs [16].

1.4

Quantum Dots and Excitons

Semiconductor Quantum Dots

Quantum Dots (QDs), sometimes referred as artificial atoms, is the object of this thesis work. It can be seen as semiconductor device with highly controlled artificial structure. Because the width of band gap is sensitive to the atomic composition of the crystal, it is possible to engineer on the width of band gap in 3-dimensional space.

In principle, the QDs are formed by the heterostructure of materials with different band gaps, which introduce a locally deformed term, VQD(r), to the effective crystal potential in Equation (1.3). The deformed effective potential not only confines the charge carriers, which become localized in QD, but also attracts the other near by free carriers and further traps them in the dot, as shown in Figure 1.5. Therefore, from quantum mechanical point of view, QDs can be seen as a potential trap for charge carriers in the crystal. It is important to mention that the QD confinement potential, in principle, is depending on the structure of QD. However, the structural dependence is not shown in the Figure 1.5 for the clear illustration of the carrier trapping mechanism.

Excitons in Quantum Dots

When one electron is excited across the band gab, one electron in conduction band and one hole in valence band are created. Since electron and hole have opposite charges, they attract each other via Coulomb force. Therefore the electron-hole pair, so called exciton, is formed. This excitation process can be seen as the creation of exciton with an annihilated photon. More importantly, the annihilation of exciton, or the recombination of electron in conduction band and hole in valence band, release the energy of exciton. The energy released by emitting a photon, abbreviated as optical transition, is of main interest in the present thesis. Other forms of the released energies will not be considered.

Excitons trapped in QDs become quantum confined. Moreover, few individual electrons and holes can also be trapped in the QDs, which resulted in various exciton complexes in QDs. Several typical few-particle exciton complexes are shown in the Figure 1.6. They are

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exci-Chapter 1. Introduction to Semiconductors and Quantum Dots Material I Material II trapped Eg,II Eg,I γ exciton creation

Figure 1.5: Schematically illustrates the origin of quantum confinement and the charge carrier trapping or creation mechanism. The structure dependence is not shown in the figure.

ton (X), negatively charged exciton (X−), double negatively charge exciton (X2−), positively charge exciton (X+), and biexciton (2X). Note that the spacial dependence of QD confinement potential is shown in Figure 1.6.

Eg CB VB X X- X2- X+ 2X (a) (b) (c) (d) (e) potentials VeQD VhQD

Figure 1.6: Schematically illustrate some few-particle exciton complexes. Spin configurations are illustrated by triangles. Only ground state and first excited state are considered. (a)One exciton state |Xi. (b)2e1h state |X−i. (c)3e1h state |X2−i. (d)1e2h state |X+i. (e)Two

excitons state |2Xi [3].

The optical properties of exciton in QDs can be determined by the emission intensity (see section 2.1.2). The excitons can be classified into two types, optically active (bright) excitons and optically inactive (dark) excitons. Note that only bright excitons can be created via photon absorption.

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Chapter 2

Single- and Few-Particle States

This section establishes the basis of the thesis work. In order to take many particles into account, we need to consider the few-particle states, which are constructed from the single-particle states. We start with the single-single-particle states, namely the effective mass approximation for quantum dots and the optical transition. Then we introduce the extension of single-particle wave function to few-particle wave function, including the concept of configuration interaction, second quantization formalism, and the dipole operator for exciton radiative decay.

2.1

Single-Particle States

2.1.1

Effective Mass Approximation for Quantum Dots

Since the charged carriers in the QDs are confined, the wave function of charge carriers are multiplied by an localized envelope function. It is sufficient to treat those envelope functions as localized quasi particles.

Effective masses are defined to describe the response of charge carriers in solid under external electric field [14]. Because these charge carriers at the edge of band behave similar compared to free electron. Effective masses are defined as, by Equations (1.5) and (1.1), m1∗ =

1 ~2∇

2 kE(k), for both electron and hole.

Mathematically, QDs are described by the locally deformed effective potential, which can be written as

VcryQD(r) = Vcry(r) + VQD(r),

where VQD is the confinement potential of QD. Therefore the periodic Hamiltonian in Equation (1.3) becomes

ˆ

Hψ(r) = ˆHper+ VQD(r) 

ψ(r) = Eψ(r). (2.1)

Using the definition of effective masses and the QD confinement potential, the excited electron and hole states in QDs can be described by effective Schr¨odinger equation,

for electron: − ~2 2m∗ e∇ 2+ VQD e (r)  χe(r) = (E − Eg) χe(r), (2.2a) for hole: − ~2 2m∗h∇ 2+ VQD h (r)  χh(r) = Eχh(r), (2.2b) where χe (χh) and VeQD (V QD

h ) are the envelope function and confinement potential of the excited electron (hole) in QDs, (see Appendix A.1). Note that for the wave function in QDs, quantum numbers (n, l, m) become good quantum numbers again, back to the notation of

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Chapter 2. Single- and Few-Particle States

atomic physics. It is because the excited states in QDs are confined in 3 dimensions, anology to electrons confined around the core of an atom. It is important to point out that only one band in considered using Equation (2.2), because single-band structure is assumed in the derivation in Appendix A.1.

2.1.2

Optical Transitions

In the electron picture, the de-excitation of a excited electron in the conduction band release the energy1. The theory of this state transition from initial state |ii to final state |f i is well developed and wildly used in various field of physics (see Appendix A.2).

In principle, we consider the interaction between the oscillating electromagnetic fields and the quantum state, where the electromagnetic field can be treated as a perturbation ˆH0(t). This interaction is described by time-dependent Schr¨odinger equation

i~∂

∂t|i(t)i = ˆ

H(t)|i(t)i.

The Hamiltonian is the sum of unperturbed system, ˆH0, and the perturbation term, ˆH0(t), namely, ˆH(t) = ˆH0+ ˆH0(t). This Hamiltonian allows us to calculate the probability, If i, of the transition from |ii to |f i by

If i= |hf | ˆH0(t)|ii|2, (2.3)

where the time dependence of states are omitted. This probability can be interpreted as the emission or absorption intensity.

2.2

Few-Particle States

2.2.1

Notations for Few-Particle System

To begin with, we introduce some notations for the following discussions. The lowercase Greek alphabets are used to describe single-particle states, |ψi or |φi, while the capital Greek alphabets are used to describe the few-particle wave functions , |ΨiN = Ψ(r1, r2, · · · , rN) or |ΦiN = Φ(r1, r2, · · · , rN), where the subscript N denote the number of particles of the system. Similar to Hilbert space for single-particle wave functions, few-particle wave function belongs to Fock space2. Further more, Φ is the Slater determinant generated by single-particle states, namely

Φ(r1, r2, · · · , rN) = φ1(r1) φ1(r2) · · · φ1(rN) φ2(r1) φ2(r2) · · · φ2(rN) .. . ... . .. ... φN(r1) φN(r2) · · · φN(rN) . (2.4)

We call |ΦiN as noninteracting state because it is constructed directly from N single-particle states, as in Equation (2.4), neglecting all particle interactions. In addition, we use |ΨiN to denote the eigenstate of few-particle Hamiltonian (Equation (2.9)), which is also referred to as the interacting state.

1It is annihilation of exciton using hole picture

2Fock space F is the function space for many particles wave functions. Mathematically, it can be defined by

tensor product of single-particle Hilbert spaces H, Fν(H) = ∞

L

n=0

SνH⊗n, where Sν represents the symmetry or

antisymmetry of each Hilbert space. It was first defined by V. A. Fock in his German paper, Konfigurationsraum und zweite Quantelung [17].

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2.2 Few-Particle States

2.2.2

Configuration Interaction

Configuration interaction (CI) is one of the oldest approach to many body problem in quantum chemistry. It is rather straight forward and perfectly suitable for few-particle cases. Since the noninteracting few-particle wave functions, Φi(r1, r2, · · · , rN), form a complete basis of Fock space, the interacting few-particle wave function, ΨCI(r1, r2, · · · , rN), can be expanded in terms of this basis,

|ΨCIi = a1|Φ1i + a2|Φ2i + · · · = X

i=1

ai|Φii, (2.5)

where |Φii is the ith lowest energy state. By minimizing the energy of CI state, hΨCI| ˆH|ΨCIi, under the constraint that CI state is normalized, hΨCI|ΨCIi = N , the coefficients aiof Equation (2.5) can be determined. It can be done by the technique of Lagrange multiplier (see Appendix A.3).

A very useful conclusion from Lagrange minimization procedure of CI is that we have the equation in the form of Schr¨odinger equation, that is

ˆ

HN|ΨCIi = E|ΨCIi. (2.6)

Note that E is the Lagrange multiplier, which is also the energy of |ΨCIi.

2.2.3

Second Quantization Formalism

Instead of using spacial variables of N particles for Slater determinant wave function, namely Ψ(r1, r2, · · · , rN), it can be represented in a more compact form with second quantization formalism using occupation number,

|ΦiN = |n1, n2, · · · , nNi, (2.7)

where ni = {0, 1} is the number of particles in the ith single-particle state |φii. The subscripts N in Equation (2.7) denote the N -particle wave function. Further more, antisymmetric nature of electrons (holes) is taken cared of by the creation and the annihilation operators, ˆe†i and ˆei (ˆh†i and ˆhi). The notation for vacuum state is |0i = |0, 0, · · · , 0i (see appendix A.4). From now on, subscript N for N-particle space will be omitted, for simplicity, through out the rest of this thesis.

Figure 1.6 shows some configurations in the QD confinement potential. Second quantization notation can be used to express the configuration in QDs. For example, states in Figure 1.6 can be written as |Xi = |1,↑ ↓ 0, ↑ 0, ↓ 0ie⊗ |0, 1, 0, 0ih = ˆe † 1ˆh † 1(|0ie⊗ |0ih), |X−i = |1, 1, 0, 0ie⊗ |0, 1, 0, 0ih = ˆe † 2eˆ † 1ˆh † 1(|0ie⊗ |0ih), |X2−i = |1, 1, 1, 0i e⊗ |1, 0, 0, 0ih = ˆe † 3eˆ † 2eˆ † 1hˆ † 1(|0ie⊗ |0ih), |X+i = |0, 1, 0, 0i e⊗ |1, 1, 0, 0ih = ˆe † 2ˆh † 2ˆh † 1(|0ie⊗ |0ih), |2Xi = |1, 1, 0, 0ie⊗ |1, 1, 0, 0ih = ˆe†2eˆ † 1ˆh † 2hˆ † 1(|0ie⊗ |0ih),

where the arrows represent the spin and the subscripts e and h denote the Fock space for electrons and holes respectively.

One useful result from second quantization formalism is that one- and two-particle operators can be elegantly calculated. As shown in appendix A.4, one- and two-particle Hamiltonians for

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Chapter 2. Single- and Few-Particle States

electrons can be written as ˆ H1−particle = X i,j Hij ˆe † ieˆj, (2.8a) ˆ H2−particle = X i,j,k,l Hijkleˆ † ieˆ † jˆekeˆl, (2.8b)

where Hij = hφi| ˆH|φji, Hijkl = hφiφj| ˆH|φkφli, and ˆe †

i|0i = |φii. Note that the spin vector is implicitly included using this notation. This result can be used to express few-particle Hamiltonian in next section.

2.2.4

Few-Particle Hamiltonian

By introducing Coulomb term, HC to Equations (2.2), one may obtain the Hamiltonian of few-particle system [18] ˆ HN = X i=1 ˆ H0(ri) + 1 2 X i6=j ˆ HC(ri, rj), where ˆ H0(r) = − ~ 2 2m∗∇ 2+ V QD(r), ˆ HC(ri, rj) =  1 4π0  qiqj |ri− rj| ,

and qi is the charge of ith particle. Note that  is the dielectric function of material, ( = 13.6 for GaAs). One thing to point out is that ˆHC is a two-particle operator. Using the hole picture with the second quantization formalism, the Hamiltonian becomes [8]

ˆ HN = X i Eieeˆ†iˆei+ X i Eihhˆ†iˆhi− X ijkl Vijkleheˆ†ihˆ†jˆhkeˆl + 1 2 X ijkl Vijklee eˆ†ieˆ†jˆekeˆl+ 1 2 X ijkl Vijklhh ˆh†iˆh†jˆhkˆhl. (2.9)

The first two terms are the single-particle energies of each electron and hole. The third term is the electron-hole Coulomb attraction energies, which runs through each electron-hole pairs. The last two terms are the electron-electron and hole-hole Coulomb repulsion energies. The Coulomb matrix elements in Equation (2.9) are

Vijkleh = hφei φhk| ˆVC|φhj φ e li =  1 4π0  Z Z φe∗ i (r) φh∗k (r0) φhj(r0) φel(r) |r − r0| d 3r d3r0 , Vijklee = hφei φej| ˆVC|φekφ e li =  1 4π0  Z Z φe∗ i (r) φe∗j (r0) φek(r 0) φe l(r) |r − r0| d 3r d3r0, Vijklhh = hφhl φhk| ˆVC|φhj φ h ii =  1 4π0  Z Z φh∗ l (r) φh∗k (r 0) φh j(r 0) φh i(r) |r − r0| d 3r d3r0 , (2.10) where VC =  1 4π0  1

|r−r0| due to the Coulomb interaction, and φi is the ith single-particle state. These matrix elements in Equations (2.10) can be calculated by Fourier convolution [19], as shown in Appendix C.1. It can be integrated analytically for the 2D harmonic oscillator QD model. Comparisons between analytical and numerical results are included in appendix C.

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2.2 Few-Particle States

2.2.5

Dipole Operator

In general, the intensity of the exciton radiative decay, Equation (2.3), can be solved by inte-grating a differential equation and the solution is depending on the polarization of the light. However, the case become much simpler under single-band assumption. We can approximate the intensity of radiative exciton decay from jth single-particle states by

Ij ≈ C Z χ∗e,j(r)χh,j(r)dr 2 ,

where C is some constant, χe,j(r) and χh,j(r) are the envelope function of recombining elec-tron and hole in jth single-particle state. Detailed derivation can be found in Appendix A.2. Whether the transition is bright or dark is determined by this transition intensity and the spin arrangement of the exciton.

Using the second quantization formalism, it is convenient to describe the intensity of few-particle radiative decay by dipole operator. Dipole operator can be defined as

ˆ

P =X

j,σ

Sjˆej,σˆhj,−σ,

which operates on any few-particle state. The summation index j runs through all the single-particle states while the index σ denote the spin. Note that only opposite spin of electron and hole can recombine. The factor Sj is defined as the jth envelope functions integration

Sj = Z

χ∗e,j(r)χh,j(r)dr.

For the noninteracting state |Φi, hΨ| ˆP |Φi adds up the transition amplitudes of all the allowed radiative exciton decays of each electron-hole pair. Therefore, we can write the intensity of the transition form a initial CI state, |Ψii =P cj,i|Φj,ii, to a final CI state |Ψfi =P cj,f|Φj,fi as

If i = |hΨf| ˆP |Ψii|2 ∝ P j,j0 cj0,f cj,ij0,f| ˆP |Φj,ii 2 ,

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Chapter 3

QD Models and Symmetries

This chapter describes the calculation models and the symmetries that is usually assumed for the QDs. There are two models used in the present thesis work, 2D model and 3D model. We start with the introduction of 2D single-band model and 3D single-band model. Then, we discuss the two types of symmetries that we focused on in this thesis work. Most of the efforts were made on 2D single-band model. However, in order to achieve better agreement with experimental results, 3D single-band model was used.

Figure 3.1: Potential profile of 2D single-band model. (a) confinement potential VQDe (x, y) of electron. (b) confinement potential VQDh (x, y) of hole. Resolution for both confinement potentials are 1.5 nm.

3.1

2D Single-Band Model

2D single-band model, abbreviated as 2D model, is a rather general model. It can be applied to various QDs system and has been wildly used [8, 9, 20, 21, 22]. For the present purpose, only s-shell and p-shell are taken into 2D model calculations. The computational resources required by 2D model were relatively small and 2D model is easy to be applied. These reasons make 2D model an ideal tool to gain some insight of the few-particle system in QDs.

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3.1 2D Single-Band Model

means that the harmonic approximation can be applied. Moreover, confinement in z-direction is stronger than in x- or y-directions, as shown in section 3.2 (QD size in z-direction is 6 nm while in x- and y-directions are 24 nm). Therefore, the confinement potential of holes and electrons can be approximated by 2D harmonic oscillators. However, the energy splitting, ~ω, of the oscillator1, needs to be fitted either by experimental data or by the symmetry requirements.

Figure 3.2: Single-particle states of electron and hole of 2D-QD Calculation. The notations of basis for each orbital are included. Only spin up states are shown (a) electron s orbital |se ↑i = |1, 0, 0, 0, 0, 0i

e (b) electron px orbital |pex ↑i = |0, 0, 1, 0, 0, 0ie (c) electron py orbital

|pe

y ↑i = |0, 0, 0, 0, 1, 0ie (d) hole s orbital |sh ↑i = |1, 0, 0, 0, 0, 0ih (e) hole px orbital |phx ↑i =

|0, 0, 1, 0, 0, 0ih (f) hole py orbital |phy ↑i = |0, 0, 0, 0, 1, 0ih . Potential Profile

The confinement potentials for electrons and holes are determined from the experimental data of energy levels. Essentially, the approxcimated 2D harmonic oscillator can be written as V (x, y) = mω22(x2 + y2), where ~ω is the separation of energy levels and m is the effective masses of confined particles. For the present purpose, we have



~ωe= 35 meV , m∗e = 0.067 me, ~ωh = 10 meV , m∗h = 0.103 me.

(3.1) Therefore by fitting the experimental data, the confinement potentials become

 VQD

e (x, y) = 5.387 × 10−4(x2+ y2) eV, VhQD(x, y) = 6.757 × 10−5(x2+ y2) eV,

1Sometimes effective length, l ef f =

p

~/mω, of the oscillator is used, which is a quantity to describe the lateral confinement.

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Chapter 3. QD Models and Symmetries

where x and y are given in the units of nanometers. Note that these potentials describe the spacial deformations at the top of valence band and the bottom of conduction band. These potentials are visualized in Figure 3.1. It is clearly shown that the confinement potential for electrons is stronger than the one for holes.

Single-Particle States

First three single-particle states, |si, |pxi, and |pyi, are taken into CI calculation. The numerical results of these three states are shown in Fiugre 3.2. Because the confinement potential for electron is stronger, electron is strongly confined while hole is weakly confined. The energy of these states from the numerical calculation are

   Ese = 34.86 meV, Esh = 9.98 meV, Ee px = 96.56 meV, Epxh = 19.94 meV, Ee py = 96.56 meV, Epyh = 19.94 meV, (3.2)

It is important to note that the diameter of hole wave functions is about 30 nm, which is too large. The large holes are the consequences of small separation energy value. However, it is still worth to use it as the first approximation.

Allowed Exciton Recombinations

It is worth to point out that the wave function integration of the single-particle state, hχe|χhi, is none zero only if the electron and the hole are in the same orbital. This means that the only possible radiative decays are

hse|shi, hpe

x|phxi, hpey|phyi.

3.2

3D Single-Band Model

In order to achieve better agreement with available experimental results, a 3D single-band model (3D model for short) is implemented. For this reason, we focus on the spectra of those few-particle configurations, which are more comparable with experiments. One thing to be noted is that 3D model is specifically fitted with the pyramidal QDs system, which makes 3D model not as general as 2D model. Moreover, it is necessary to take the structure of pyramidal QDs into considerations in order to set up a reasonable 3D model. On the other hand, when carry a 3D model calculation, fit to the experimental data is not needed. The energy separation is uniquely determined by QD structure.

Structure of Pyramidal QD System

In general, self-limited QD located at the top of pyramid due to the thickness variation. In addition to the QD, there are two vertical quantum wires (VQWRs) connected to the top and the bottom of the QD. The formation of these quantum structure is attributed to the epitaxial growth during the fabrication procedure.

Detailed pyramid structure information has been obtained from the work done by Zhu et al [23]. For the pyramid used in the present thesis work, the concentration of indium decreased from 20% at the center to 10% at the edge in the QD region (shown in Figure 3.3, region I). We model the size of the QD region as 24 nm wide and 6 nm height; the band gap increases quadratically with the concentration as we go farther from the center. At VQWRs regions (region II), the concentration of aluminum is constant 5%. We model this region as constant potential cylinder with radious 16 nm. At backgound region (region III) we use the constant band gap 1.923 eV for AlxGa1−xAs.

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3.2 3D Single-Band Model x y z QD VQWR VQWR I : QD II : VQWRs 16 nm 24 nm 6 nm

III : AlGaAs background

Figure 3.3: Schematic illustration of the QD structure and the 3D model. Three regions are shown by the arrows.

Hole Structure

It is important to note that the pyramidal QDs is grown on the [111] direction (see section 1.3) of GaAs crystal, which is aligned with z-direction. Therefore the heavy hole effective mass is heavier in z-direction than in x- and y-directions. It can be derived from k · p theory that

m∗hh,z = γ1−2γ31 , and m∗hh,⊥ = γ1+γ21 ,

where γ1 = 6.85, γ2 = 2.1, and γ3 = 2.9 [24], are the so called Luttinger parameters, for GaAs. mhh,⊥ is the effective mass in any direction in xy plane. Put the values into the formula of effective mass, we have

m∗e = 0.067 me, m∗hh,z = 1.282 me, m∗hh,⊥ = 0.106 me.

Note that heavy hole in z-direction is more than 10 times heavier than in x- and y-directions. Potential Profile 40 nm 40 nm x z y 24 nm 6 nm QD VQWR 16 nm VQWR 40 n m

(a) 3D potential grid

CB QD VQWRs Eg=1.345 eV VB (b) potential profile

Figure 3.4: Visualization of 3D confinement potentials. (a) the 3D numerical grid with potential isosurface. (b) schematically illustrates the potentials with different regions.

3D model is based on the idealized spacial structure of the QD. As in section 3.2, the cladding VQWRs and AlxGa1−xAs background are needed to be considered. To simplify the problem,

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Chapter 3. QD Models and Symmetries

we assume that the confinement potentials are constant except for QD region. Moreover, we neglect the concentration variation in z-direction, meaning that QD confinement potentials can be written is the following form

V (x, y, z) = 1 2mω

2(x2+ y2) + C,

for some constant C = V (0, 0, z). Together with the experimental data, we set our 3D confine-ment potential as Ve(r, z) =      0 r 2 → 30, r < 12, |z| ≤ 3. QD 115, r < 8, |z| > 3. VQWRs

331, else where. background

, (3.3)

for electrons and

Vh(r, z) =      0→ 50, r < 12, |z| ≤ 3. QDr2 125, r < 8, |z| > 3. VQWRs

247, else where. background

, (3.4)

for holes, where r = px2+ y2. The energy and length units in Equation (3.3) and (3.4) are given in meV and nm respectively. Note that the band gap energy, Eg = 1.345 eV, is subtracted from these potentials in order to have zero energy for both electrons and holes when (x, y, z) = (0, 0, 0). These confinement potentials can be visualized in Figure 3.4(b).

Single-Particle States

One thing that is needed to be considered is the atomic p-orbital nature of holes, which leads to heavier hole effective mass in [1 1 1] crystal direction. This non isotropic effective mass gives rise to pz-orbital of the envelope function as the first excited hole state, Figure 3.6, while the corresponding first excited electron states remain in x- and y-directions, Figure 3.5.

Figure 3.5: Numerical results of electron orbitals isosurface (a) s electron, (b) px electron and

(c) py electron are degenerated first excited states.

The calculated energies of these states for the symmetric QD are    Ese = 72.588 meV, Ee px = 109.510 meV, Ee py = 109.510 meV. and  E h s = 28.294 meV, Epzh = 43.227 meV.

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3.3 Symmetries of Quantum Dots

Figure 3.6: Numerical results of hole orbitals isosurface. (a) s hole, (b) first excited pz hole. Allowed Exciton Recombinations

One thing that makes 3D model significantly different from 2D model is that none of the electron wave functions overlap with the hole in pz orbital, meaning that hχe|χh,pzi = 0 for s, px, and py orbitals. Therefore the only allowed optical transitions involve the recombination of a hole in the s-shell.

3.3

Symmetries of Quantum Dots

Since the asymmetry effects are the objects of the present thesis work, it is necessary to in-troduce the symmetries that we focused on. Among the works that devoted to the spectra calculations, two symmetries are usually assumed.

Hidden Symmetry

The first is the symmetry between electron and hole, which resulted in identical single-particle states. We call this symmetry as hidden symmetry, which is also used by other authors [10]. The consequence of hidden symmetry is the symmetric Coulomb matrix elements, namely Vijklhh = Vklijee and Vijkleh = Vljkiee , meaning that hole-hole and electron-hole interactions are re-lated to electron-electron interaction. This relation simplify the emission spectra significantly because of the cancellations between these interactions.

Rotational Symmetry

The second symmetry is the symmetry in different directions, namely, the rotational symmetry. Under this symmetry we have the degenerate excited states and the angular momentum is conserved. We expect to get more peaks when the rotational symmetry breaks down, due to the splitting of those degenerated excited states.

Level of Symmetry

We expect both of these symmetries (hidden symmetry and rotational symmetry) to be slightly broken for real QDs. To compare with the experimental results, we remove these symmetries one by one. Firstly, we remove the hidden symmetry, in order to fit the energy levels measured from the experiments. Then, we break the rotational symmetry, in order to achieve better agreement with experiment. Note that for 3D model, we discuss only rotational symmetry

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Chapter 3. QD Models and Symmetries

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Chapter 4

Results and Discussions

We discuss all the results from the calculation in this chapter. The results can be separated into two parts. The first part is the asymmetry effects on neutral exciton complexes, namely 4X, 5X, and 6X. Moreover, Auger-like satellite of 2X spectra is discussed. The second part is the charged exciton complexes, X+, X−, X2−, X3−, X4−, and X5−, which are compared with experimental data.

Unless otherwise noted, all the emission spectra are generated by the transition from exci-tonic ground state

n e, m h → (n − 1)e, (m − 1)h,

where n and m denote the number of electrons and holes respectively. Note that ground state spectra means that the temperature is absolute zero.

4.1

Notations and Concepts

Before we start, it is worth to introduce some of the notations and concepts that are used for this section.

Hamiltonian

In general, if we use the noninteracting states (see section 2.2.1) to express the few-particle Hamiltonian, it can be divided into two parts. The diagonal part and the nondiagonal part, namely

ˆ

H = ˆHd+ ˆHn.

Note that each of the noninteracting state, |Φi, is a eigenstate of the diagonal part of the Hamiltonian. We further introduce the eigenvalues of the diagonal Hamiltonian as

ˆ

Hd|Φi = Hd

Φ|Φi, (4.1)

where |Φi can be any noninteracting state and Hd

Φis the related eigenvalue. Note that HΦd, which is the summation of all possible energy terms, can be interpreted as the energy of noninteracting state |Φi. For example, HXd = Ese+ Esh− Veh

ss,d for noninteracting exciton ground state |Xi. Furthermore, we call the truncated Hamiltonian that we used in the 2D and 3D model as full Hamiltonian1 in order to introduce the simplified Hamiltonian, which is used to gain some intuitive understanding of the few-particle system.

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Chapter 4. Results and Discussions

Scattering between States

The Coulomb terms in Equation (2.9) allow the particles to be scattered from one state to another state, e.g. ˆVeh

ijkl = Vijkleh ˆe † iˆh

jˆhkˆel moves lth electron and kth hole to ith position and jth position respectively. This scattering conserves the total spin of electrons and holes separately because the Coulomb matrix elements are zeros for opposite spin in Equation (2.10). For example, ˆ V6512eh |se ↑, pe x ↑ie⊗ |sh ↓, phx ↓ih = ˆV6512eh | ↑ 1, ↓ 0 |{z} s , 1,↑ ↓ 0 |{z} px , 0,↑ ↓ 0 |{z} py ie⊗ | 0, 1 |{z} s , 0, 1 |{z} px , 0, 0 |{z} py ih = |0, 0, 1, 0, 1, 0ie⊗ |0, 0, 0, 1, 0, 1ih = |pey ↑, pe x ↑ie⊗ |pey ↓, pex↓ih, which can be seen as ˆVeh

6512 scatters exciton from s-orbital into py-orbital. Note that all the scattering terms of the Hamiltonian are belonging to the nondiagonal part of the few-particle Hamiltonian.

Coulomb Matrix Elements

Because the Coulomb matrix elements in Equation (2.9) are identical for spin up and spin down, we use the simplified notation for Veh

ijkl

Direct interaction: Vbb,dhh = V4334hh = Vbbbbhh, Direct interaction: Vab,dhh = V1441hh = Vabbahh , Exchange interaction: Vee ab,x = V2424ee = Vababee , Exchange interaction: Vee ac,x = V1515ee = Vacacee , Scattering: Vee ba,s = V3421ee = Vbbaaeh , Scattering: Veh aa,s = V2121eh = Vaaaaeh , .. . ... (4.2)

The subscripts a, b, and c denote the s-orbital, px-orbital, and py-orbital for 2D model, while b denotes the pz hole orbital in 3D model. The subscripts x and d denote the exchange and direct interactions, while subscript s denote the scattering terms of the Hamiltonian. Note that the spin orientation defines whether the interaction is direct of exchange. The up scripts ee, hh, and eh denote the electron-electron, hole-hole, and electron-hole interactions respectively. The Coulomb matrix elements are listed in table C.2. We use this simplified notations though out the rest of this thesis, unless otherwise noted.

4.2

Neutral Excitons

4.2.1

Breaking of Hidden Symmetry

It is worth to compare the results with the previous work done by Bare et al [9]. The emission spectra of the symmetric QD in 2D model are shown in Figure 4.1. 4.1(a) shows the spectra with fitted energy levels with experiment (see section 3.1), while 4.1(b) is generated by input the same Coulomb matrix elements as in Bare’s work. Note that the spectra in Figure 4.1(b) perfectly reproduce the one reported by Bare et al (see reference [9] figure 1). The energies are measured from the band gap. The peaks are labeled by total spin of final state for electrons and holes, as labeled in the reference [9].

In general, the emission spectra in Figure 4.1(a) can be divided into two groups. Higher emission energy (more than 70 meV) for p-shell exciton recombinations and lower emission

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4.2 Neutral Excitons

0 20 40 60 80

Emission Energy (meV)

Intensity (arbitrary unit)

0 20 40 60 80 4X to 3X 5X to 4X 3X to 2X 6X to 5X 2X to 1X 1X to 0X 4X to 3X 5X to 4X 3X to 2X 6X to 5X 2X to 1X 1X to 0X (a) (b) ss dd ss ss tt st/ts ss dd dd dd ss st/ts tt tt ss dd dd ss

Emission Energy (meV) ss dd ss dd dd ss ss ss tt tt ss ss stts ts st dd dd dd dd dd dd dd dd

Figure 4.1: Numerical result of spectra with different number of excitons δ-function are replaced by narrow Lorentzians for better visibility. (a) shows the spectra from the fitted energy level. (b) shows the reproduced spectra from the parameters from Bare at el.

energy (less than 40 meV) for s-shell exciton recombinations. From the first sight, the fit-ted results (Figure 4.1(a)) are quit different from the previous study (Figure 4.1(b)). First, the 4X spectrum seems to be totally different. Besides, instead of stable p-shell exciton re-combination energy for different number of particles in 4.1(b), the spectra in Figure 4.1(a) shows increasing p-shell exciton recombination energies versus number of particles. Finally, the triplet-singlet/singlet-triplet peaks in 3X and 5X spectra split, as illustrated in Figure 4.1(a) by circles.

These differences can be attributed to the breaking of the hidden symmetry. The hid-den symmetry refers to the ihid-dentical envelope function of electron and hole, χe and χh, as well as degenerated single particle states (see section 3.3). The Coulomb interactions are then symmetric for electrons and holes. Therefore few-particle energy of a specific state can be dramatically simplified, due to cancellations of Coulomb energies. For example, be-cause the exchange interactions for electrons and holes are symmetric, the energy of 2X triplet-singlet state, |tsi = |1, 0, 1, 0, 0, 0ie⊗ |1, 1, 0, 0, 0, 0ih, is identical2 to singlet-triplet state, |sti = |1, 1, 0, 0, 0, 0ie⊗ |1, 0, 1, 0, 0, 0ih, under such symmetry. The notation for singlet-triplet is defined as electrons are singlet while holes are triplet as follow

|single −e h tripleti = |sti = | se ↑, se | {z } singlet i ⊗ | sh ↑, ph x ↑ | {z } triplet i = | 1,↑ ↓ 1 |{z} s , 0, 0 |{z} px , 0, 0 |{z} py ie⊗ | ↑ 1, 0 |{z} s , 1, 0↑ |{z} px , 0, 0 |{z} py ih.

The arrows represent the spin up and spin down state for each particle.

In order to fit with experimental data, confinement potentials of electrons and holes must be

2The condition for E

ts= Est is that Epe− Vab,xee = E h p − Vab,xhh

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Chapter 4. Results and Discussions r = 1.079 r = 1.133 r = 1.206 r = 1.311 r = 1.491

Intensity (arbitrary unit)

0 20 40 60 80 100 120 0 (c) 5X spectra 20 40 60 80 100 120 (a) 3X spectra 0 20 40 60 80 100 120 r = 1.036 Energy (meV) r = 1.079 r = 1.133 r = 1.206 r = 1.311 r = 1.491 r = 1.036 r = 1.079 r = 1.133 r = 1.206 r = 1.311 r = 1.491 r = 1.036

(b) 4X spectra sudden switch

Figure 4.2: Illustration of spectra evolve with hole confinement potential. The r value is the ratio between s-shell direct Coulomb matrix element, namely r = Vaa,dee /Vaa,dhh . The fitted values used in the 2D model is r = 1.491. The split peaks are emphasized by dashed circles in 3X and 5X spectra. There is a sudden switch of spectra feature in 4X spectra (a) evolution of 3X spectra (b) evolution of 4X spectra (c) evolution of 5X spectra.

adjusted. As shown in Figure 3.2, the envelope function of electrons and holes are significantly different, which breaks the hidden symmetry. However, it is possible to engineer on the confine-ment potentials numerically to regain such symmetry. By increasing the confineconfine-ment energy of holes, from ~ωh = 10 meV to ~ωh = 24.7 meV, we get the ratio r = Vaa,dhh/Vaa,dee ≈ 1. It is clearly shown in Figure 4.2 that the the symmetry breaking of Coulomb matrix elements evolves the spectra from Bare’s work (Figure 4.1(b)) to the results fitted with experimental data (Figure 4.1(a)). Specifically, the triplet-singlet/singlet-triplet peak in 3X and 5X spectra splits into two peaks, triplet-singlet peak and singlet-triplet peak, because Ee

p− Vab,xee 6= Eph− Vab,xhh. The shift of the p-shell exciton recombination energy can be explained by this breaking of hidden symmetry as well. As shown in Figure 4.2, 5X p-shell exciton recombination peak evolves from 93 meV to 80 meV, while 3X p-shell exciton recombination peak evolves from 93 meV to 75 meV. This difference makes 5X p-shell exciton recombination peak having 5 meV higher energy than 3X p-shell exciton recombination peak. It is due to that there are more carrier-carrier interactions in the 5X spectra than in the 3X spectra. These interactions cancel each other by the presence of hidden symmetry, which gives rise to the stable p-shell exciton recombination energy under hidden symmetry.

For 4X spectra, there is a sudden switch from the Bare’s spectra to the one fitted with experiment; we postpone the explanation to later analysis in next section.

4.2.2

Breaking of Rotational Asymmetry

The rotational asymmetry the of QD is introduced to the confinement potential by asymmetry factor ay VQD(x, y, ay) = m∗ω2 2 (x 2+ a yy2). (4.3)

Because asymmetry is considered as a structural defect, value of ay is the same for both electrons and holes. Some numerical results are shown in Appendix C. Figure 4.4 illustrates how the |pxi and |pyi split.

The spectra of 4X, 5X, and 6X are shown in Figure 4.3. Two features can be observed when asymmetry factor becomes larger. The first is that the blue shift of each peak linearly

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4.2 Neutral Excitons

0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100

Intensity (arbitrary unit)

Energy (meV) ay=1.00 ay=1.05 ay=1.09 ay=1.13 ay=1.00 ay=1.05 ay=1.09 ay=1.13 ay=1.00 ay=1.05 ay=1.09 ay=1.13

(a) 4X spectra (b) 5X spectra (c) 6X spectra

sudden switch

Figure 4.3: Numerical results of asymmetric QD spectra of 4X, 5X, and 6X. ay is the

asym-metry factor, which grows from 1 to 1.13. The asymasym-metry peaks due to rotational asymasym-metry are illustrated by the arrows. (a) 4X spectra, there is a sudden switch, as in the discussion of hidden symmetry, when ay > 1.09. (b) 5X spectra, there are paired asymmetry peaks. Not only

the intensities but also the splitting separation increased with asymmetry factor (c) 6X spectra, the intensity of asymmetry peak increased with asymmetry factor

depends on the asymmetry factor. The second is some extra “asymmetry peaks” at p-shell recombinations region show up, illustrated by the tilted arrows in Figures 4.3. These two features need to be explained separately. The reason for blue shift of the peaks is trivial. Because the asymmetry factor adds to the confinement potential, the quantum energy level in y-direction is increased by multiplying the asymmetry factor. As a result, the energy increment is linearly depending on asymmetry factor, as listed in Appendix C.3. However, the explanations for asymmetry peaks are less trivial. We discuss the asymmetry peaks case by case in the following sections

6X Spectra: Increasing Intensity

To explain the asymmetry peaks for the asymmetric QDs, let us focus on the simplest case, 6X → 5X spectrum. There are only 3 final noninteracting states for (e ↓, h ↑) recombination3

|si = |1, 0, 1, 1, 1, 1ie⊗ |0, 1, 1, 1, 1, 1ih = |ai, |pxi = |1, 1, 1, 0, 1, 1ie⊗ |1, 1, 0, 1, 1, 1ih = |bi, |pyi = |1, 1, 1, 1, 1, 0ie⊗ |1, 1, 1, 1, 0, 1ih = |ci,

(4.4)

which can be used as a restricted basis for the 5X Hamiltonian. s (px and py) on the left hand side represents the s-orbital (px-orbital and py-orbital) recombination.

For 6X case, there are three kinds of scattering processes: (s, px scattering) : Vˆab,seh, (s, py scattering) : Vˆac,seh, (px, py scattering) : Vˆbc,seh.

For |bi, there are only two scattering terms needed to be considered, namely ˆVbc,xeh and ˆVba,seh.

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Chapter 4. Results and Discussions

Therefore we have4

ˆ

H|bi = ( ˆHd+ ˆHn)|bi

= Hd

b|bi − ˆVbc,seh + ˆVba,seh|bi

= Hd

b|bi − Vbc,seh|ci − Vba,seh |ai, where ˆHd|bi = Hd

b|bi as defined in Equation (4.1). Assume that we have the few-particle eigenstate |Ψi = α|ai + β|bi + γ|ci, it is possible to solve the coefficients α, β, and γ by diagonalizing the simplified Hamiltonian

ˆ

H|Ψi = (αHd

a− βVba,seh − γVca,seh)|ai + (βHbd− γVeh cb,s− αVab,seh)|bi + (γHcd− βVeh bc,s− αV eh ac,s)|ci. That is ˆH can be written as

ˆ H =   Hd a −Vba,seh −Vca,seh −Veh ab,s Hbd −Vcb,seh −Veh ac,s −Vbc,seh Hcd  , (4.5)

using Equations (4.4) as basis. Note that this Hamiltonian has been significantly simplified from the full 5X Hamiltonian, which is a 36 by 36 matrix.

We have to solve α, β, and γ numerically for the eigenstates. But, luckily, when the QD is symmetric, it is possible to derive a simple expression for the state which gives rise to the asymmetry peak. Consider the linear combination |Ψayi = β|bi − γ|ci = √12|bi −√12|ci, it is an eigenstate of the simplified Hamiltonian, because Vba,seh = Vca,seh and Hbd = Hcd for the symmetric QD5, ˆ H|Ψayi = ˆHd|Ψayi + ˆHn|Ψayi = √1 2  Hd b|bi − Hcd|ci  + √1 2  − Veh

bc,s|ci − ———Vba,seh|ai + Vccbbeh |bi + ———Vca,seh |ai  = √1 2  Hbd+ Vcb,seh  |bi − 1 2  Hcd+ Vbc,seh  |ci =Hbd+ Vbc,seh|Ψayi.

Note that the other two eigenstates turn out to be the final states of s- and p-shell exciton recom-bination peaks. The transitions to |Ψayi gives zero emission intensity because I ∝ |hΨ| ˆP |6Xi|2, where ˆP is the dipole operator for emission, |6Xi is the six exciton ground state, and |Ψi is the 5X final state. For the case of |Ψayi = √12|bi − √12|ci, we have

|hΨay| ˆP |6Xi|2 = |√12hb| ˆP |6Xi − √12hc| ˆP |6Xi|2

= 50%|hb| ˆP |6Xi|2− 50%|hc| ˆP |6Xi|2 = 0

where hb| ˆP |6Xi = hc| ˆP |6Xi, meaning that the transition to |Ψayi is forbidden. However, when the asymmetry is introduced to the QD, we have Hbd6= Hd

c and Vba,seh 6= Vca,seh. This asymmetry redistribute the weight of each noninteracting state, which contributes to a small intensity to the dark state transition. As the asymmetry factor gets larger, the weight of |bi gets heavier. For example 73.32% of |bi, 26.65% of |ci, and 0.03% of |ai, when ay = 1.05, and we have

|hΨay| ˆP |6Xi|2 ≈ 73.3%|hb| ˆP |6Xi|2− 26.7%|hc| ˆP |6Xi|2 6= 0.

4The signs of creation and annihilation operators nicely cancel each other. 5The values of Coulomb matrix elements are as shown in Table C.2

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4.2 Neutral Excitons

The eigenstates of the simplified Hamiltonian, Equation (4.5), agrees exactly with the full Hamiltonian, meaning that it is sufficient to neglect all other configurations without losing any understanding. This result can be interpreted intuitively that the asymmetry peak origins from the splitting of p-shell, as illustrated in Figure 4.4(b), but the exact position of the asymmetry peak and the intensity can not be explained by this naive picture.

px py s a b c Asymmetry peak

Figure 4.4: Schematic illustration of rotational asymmetric 6X radiative decay. Electrons are represented by gray triangles while holes are represented by blank triangles. The direction of each triangle indicates the direction of spin. Note that py orbital is the higher energy one. Breaking of

rotational symmetry is illustrated by the dashed line. The intuitive explanation for 6X radiative decay is shown by the wavy line.

5X Spectra: Splitting Peaks

The same concept, as 6X spectrum, can be used to explain the asymmetry peaks of 5X spectra. However, it is difficult to obtain a simple intuitive explanation which agrees with the full Hamiltonian. It is because the number of configurations blow up for 4X final states, which increase the number of available scattering processes. The short discussion here is focused on the splitting asymmetry peaks of 5X → 4X spectra.

As shown in Figure 4.3(b), there are paired asymmetry peaks in 5X spectra. When the asymmetry factor increases from ay = 1.05 to ay = 1.13, not only the intensities of the paired peaks but also the splitting separation increased. The splitting of 5X asymmetry peaks can, again, be attributed to the breaking of hidden symmetry, which leads to splitting of electron-electron exchange interaction and hole-hole exchange interaction. Some typical 4X final states of 5X spectra, after the recombination of a p-shell exciton, are

p-shell e-e exchange states (

|↑↑,e ↑↓i = |1, 1, 1, 0, 1, 0ih e⊗ |1, 1, 1, 0, 0, 1ih, |↑↑, ↓↑i = |1, 1, 1, 0, 1, 0ie⊗ |1, 1, 0, 1, 1, 0ih, p-shell h-h exchange states  |↑↓, ↑↑i = |1, 1, 1, 0, 0, 1ie⊗ |1, 1, 1, 0, 1, 0ih,

|↓↑, ↑↑i = |1, 1, 0, 1, 1, 0ie⊗ |1, 1, 1, 0, 1, 0ih,

where the arrows denote the direction of p-shell carrier spin. Similar to the case of 6X spectra, the linear combination

|Ψe,xi = √12 |↑↑, ↑↓i − |↑↑, ↓↑i, |Ψh,xi = √12 |↑↓, ↑↑i − |↓↑, ↑↑i,

is the final states for the asymmetry peaks. The different energy of these two states, which results in the a paired splitting asymmetry peaks, arise from the asymmetric exchange interac-tion (also see secinterac-tion 4.2.1). For example, as shown in Appendix C.3, Vee

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Chapter 4. Results and Discussions

Vhh

ccbb = 0.930 meV when ay = 1.05. But, one thing different from the 6X spectra is that the weight of |↑↑, ↑↓i and |↑↑, ↓↑i do not redistribute when the asymmetry is introduced to the QD. It is because there is no energy difference between these two states, and the scattering term cancels each other. On the other hand, the increasing intensity of 5X asymmetry peaks is contributed by the asymmetric initial 5X states6.

4X Spectra: Sudden Switch

(a) (b)

Figure 4.5: (a) shows the unpolarized 4X ground state. There are three paired electron-electron exchange interaction, while only two paired hole-hole exchange interaction. (b) shows the polarized 4X ground state, which gives rise to sudden switch of 4X spectra.

The sudden switch of 4X → 3X spectra appears when the asymmetry factor becomes ay & 1.09 in Figure 4.3 and when the ratio of Coulomb interaction becomes r & 1.491 in Figure 4.2. The reason for the sudden switch is the polarization of orbital wave function. For the weakly deformed QD, a typical 4X ground states is

|Ψweaki = |1, 1, 1, 0, 1, 0ie⊗ |1, 1, 0, 1, 0, 1ih,

as shown in Figure 4.5(a). The spin arrangement of |Φweaki is due to Hund’s law, where each orbital of a given shell is half populated by parallel spin electrons before and of the orbital is fully occupied. The Hund’s law can be explaned by the exchange interaction between electrons with parallel spins, which lower to total energy of |Φweaki. On the other hand, a typical 4X ground state of a strongly deformed QD is

|Ψstrongi = | 1, 1 |{z} s , 1, 1 |{z} px , 0, 0 |{z} py ie⊗ |1, 1, 1, 1, 0, 0ih,

as shown in Figure 4.5(b), where py is the higher energy orbital. The number of available optical transitions is dramatically decreased when the ground state transform from |Ψweaki to |Ψstrongi. This transformation happens only if the energy of |Ψstrongi becomes lower than the energy of |Ψweaki. Intuitively, if we neglect the scattering processes7, this transformation can be seen as the competition between the energy of p-shell and the energy of exchange interaction. For the diagonal energies Hd

s = hΨstrong| ˆH|Ψstrongi and Hwd = hΨweak| ˆH|Ψweaki, we have Hsd = 2E1e+ 2E2e− 2Vee ab,x + 2E1h+ 2E2h− 2Vab,xhh − Cseh Hwd = 2E1e+ E2e+ E3e− Vee ab,x− V ee ac,x− Vcb,xee  + 2Eh 1 + E2h+ E3h− Vab,xhh − Vac,xhh − Vcb,xhh + Cweh,

6For 6X spectra, the increasing intensity of asymmetry peak is contributed by the asymmetric final 5X state 7From the numerical point of view, it is reasonable to neglect the scattering terms because the weight

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4.2 Neutral Excitons

where Ceh

s and Cweh are the electron-hole Coulomb attraction energies for |Ψstrongi and |Ψweaki respectively. a, b, and c denote s-orbital, px-orbital, and py-orbital as in 6X spectra. Note that there are four paired exchange interactions for the electrons of |Ψstrongi, while there are six paired exchange interactions for the electrons of |Ψweaki. If we define ∆E = Hs− Hw, we can conclude that the sudden switch happens when ∆E change sign from ∆E ≥ 0 to ∆E ≤ 0. It is interesting to point out that as the asymmetry factor becomes larger, the 4X spectra become simpler, which is contrary to our expectation.

4.2.3

Auger-like Satellites: Biexcition

Auger-like satellite refers to the emission signal coming from the exciton radiative decay, which also excites other particle none radiatively. Intuitively, as shown in Figure 4.6, when the exciton (e ↑, h ↓) recombines a part of the energy is transfered to the other exciton (e ↓, h ↑), which is excited and jumps to higher orbital.

emitted photon excitation

excitation radiative

decay

Figure 4.6: Schematically illustration of Auger-like emission.

Since the observation of the Auger-like satellite is possible to carry out in the experiment, it is interesting to calculate these Auger-like satellite. For the few-particle configurations, only biexciton has the Auger-like satellite emission, when s and p orbital are taken into account. In addition, since the intensities of Auger-like satellite are expected to be weak, the relative position to other strong peaks is critical to find the Auger-like satellite, we focus ourself on the positions of the Auger-like satellite from the biexciton main emission.

Results from 2D Calculations

The calculated like satellite are shown in Figure 4.7(a). The separation between Auger-like satellite and 2X main emission peak is about 47.50 meV. The intensity is rather small, the intensity ratio ,Iauger/Imain, is about 7.4 × 10−5. A concrete explanation for the Auger-like satellite is due to the few-particle effects. To see this, we define some biexciton noninteracting initial states

2X ground state: |gi2X = |1, 1, 0, 0, 0, 0ie⊗ |1, 1, 0, 0, 0, 0ih 2 holes excited state: |hhi2X = |1, 1, 0, 0, 0, 0ie⊗ |0, 0, 1, 1, 0, 0ih 1 exciton excited state: |ehi2X = |1, 0, 0, 1, 0, 0ie⊗ |0, 1, 1, 0, 0, 0ih,

and we define |2XgiCI to be the CI ground state of biexciton configuration. Therefore, for symmetric QD, 2D calculation yields the biexciton CI ground state as

References

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